Problem 24 - Essential Calculus Early Transcendentals

Question

(a) Apply Newton's method to the equation \(1 / x-a=0\) to derive the following reciprocal algorithm: $$x_{n+1}=2 x_{n}-a x_{n}^{2}$$ (This algorithm enables a computer to find reciprocals without actually dividing.) (b) Use part (a) to compute 1\(/ 1.6984\) correct to six decimal places.

Step-by-Step Solution

Answer

The reciprocal of 1.6984 is approximately 0.58868.

Step 1: Understand Newton's Method

Newton's method is an iterative root-finding algorithm. For a function \( f(x) \), it uses the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). We apply this method to the equation \( \frac{1}{x} - a = 0 \).

Step 2: Identify Function and Derivative

For the equation \( \frac{1}{x} - a = 0 \), define \( f(x) = \frac{1}{x} - a \). The derivative of \( f(x) \) is \( f'(x) = -\frac{1}{x^2} \).

Step 3: Apply Newton's Formula

Substitute \( f(x) \) and \( f'(x) \) into Newton's method: \( x_{n+1} = x_n - \frac{\frac{1}{x_n} - a}{-\frac{1}{x_n^2}} \). Simplifying gives \( x_{n+1} = x_n + x_n^2(\frac{1}{x_n} - a) \), which further simplifies to: \( x_{n+1} = 2x_n - ax_n^2 \).

Step 4: Setup the Iterative Process to Compute Reciprocals

Using the derived formula \( x_{n+1} = 2x_n - ax_n^2 \), where \( a = 1.6984 \), we will iterate to find the reciprocal. Choose an initial guess, say \( x_0 = 0.6 \), which is a value close to our estimate of \( \frac{1}{1.6984} \).

Step 5: Perform Calculations for Convergence

Iteratively calculate:- \( x_1 = 2 \times 0.6 - 1.6984 \times (0.6)^2 = 0.58848 \)- \( x_2 = 2 \times 0.58848 - 1.6984 \times (0.58848)^2 \approx 0.58934 \)- Continue until convergence to a required precision (six decimal places).

Step 6: Verify Results

Check each subsequent approximation: After a few iterations, the values stabilize around \( x \approx 0.58868 \). Verify by checking that \( 1.6984 \times 0.58868 \approx 1 \).

Key Concepts

Reciprocal Algorithm

Let's dive into the exciting world of reciprocal algorithms. A reciprocal algorithm is a method used to find the reciprocal of a number, often without directly performing division. In simpler terms, it's a way to calculate the value of \( \frac{1}{a} \) for any number \( a \). This can be particularly useful for computational purposes where dividing might be expensive or slow.

Newton's method can be adapted into a reciprocal algorithm. This involves setting up an equation whose solution will provide the reciprocal. In our case, we use \( \frac{1}{x} - a = 0 \), which transforms into the iterative formula:

\[ x_{n+1} = 2x_n - ax_n^2 \]

This cleverly structured formula allows a computer to compute reciprocals through iteration rather than division. Computers are exceptionally good at handling repetitive calculations, which makes this approach efficient and speedy in practice.

Using a reciprocal algorithm is like outsourcing the job of division to a methodical step-by-step computation, allowing for increased performance in computational tasks.

Iterative Process

The term 'iterative process' describes a repetitive set of procedures aimed at getting closer to a desired outcome step by step. In mathematics, it's used to refine an approximation incrementally until it reaches an acceptable level of precision.

When dealing with algorithms like Newton's method, the iterative process is a core component. Starting from an initial guess, we apply the same formula repeatedly:

• Make an initial guess, such as \( x_0 = 0.6 \).
• Compute the next approximation using the formula \( x_{n+1} = 2x_n - ax_n^2 \).
• Repeat until the change in successive approximations is minimal.

As each iteration brings us closer to the exact reciprocal, the key to this algorithm's success lies in the ability to refine guesses quickly and accurately, making the iterative process an efficient tool in root finding and other calculations.

Root-finding Algorithm

A root-finding algorithm is designed to identify the roots of a given function—essentially, the points where the function equals zero. In this problem, we're using Newton's method to find the root of the equation \( \frac{1}{x} - a = 0 \).

This specific implementation of a root-finding algorithm functions by improving an initial guess to zero in on the true root, which here translates directly to solving for the reciprocal of \( a \). Here's how it works:

• Identify the function, \( f(x) = \frac{1}{x} - a \), whose root we need.
• Calculate the derivative, \( f'(x) = -\frac{1}{x^2} \), that's used to adjust our guesses.
• Adjust guesses using Newton's iterative formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \).

Newton's method relies on the powerful principle of linear approximation to drive each cycle closer to the exact root. It is the backbone of many numerical methods and computational techniques, serving as a bright example of mathematics aiding computation to solve problems efficiently and reliably.